Consider a function $f(z)=\cos(z)\cosh(az)+\sin(z)\sinh(bz)$ for $z\in \mathbb{Z}, a,b \in \mathbb{R}$. Denote D being a set {a,b} of such pairs of parameters that NOT ALL zeroes of corresponding $f(z)$ belong to coordinate axes in $z$. Obviously the line $b=0$ are not in D, as zeroes of $\cos(z)\cosh(az)=0$ belong to axes $Re z=0, Im z=0$.

Problem: define geometrically the set D in the plane {a,b} as exact as possible.

Calculations show that a set D is finite near the origin. How to describe it more exact? Is it possible to find a ball centred at origin for which D is inside this ball? Is it possible to find estimates for a radius of this ball? More suggestions?